Mapping numerical magnitudes onto symbols: the numerical distance effect and individual differences in children's mathematics achievement

Although it is often assumed that abilities that reflect basic numerical understanding, such as numerical comparison, are related to children’s mathematical abilities, this relationship has not been tested rigorously. In addition, the extent to which symbolic and nonsymbolic number processing play differential roles in this relationship is not yet understood. To address these questions, we collected mathematics achievement measures from 6- to 8-year-olds as well as reaction times from a numerical comparison task. Using the reaction times, we calculated the size of the numerical distance effect exhibited by each child. In a correlational analysis, we found that the individual differences in the distance effect were related to mathematics achievement but not to reading achievement. This relationship was found to be specific to symbolic numerical comparison. Implications for the role of basic numerical competency and the role of accessing numerical magnitude information from Arabic numerals for the development of mathematical skills and their impairment are discussed.     

Children’s mapping between symbolic and nonsymbolic representations of number

When children learn to count and acquire a symbolic system for representing numbers, they map these symbols onto a preexisting system involving approximate nonsymbolic representations of quantity. Little is known about this mapping process, how it develops, and its role in the performance of formal mathematics. Using a novel task to assess children’s mapping ability, we show that children can map in both directions between symbolic and nonsymbolic numerical representations and that this ability develops between 6 and 8 years of age. Moreover, we reveal that children’s mapping ability is related to their achievement on tests of school mathematics over and above the variance accounted for by standard symbolic and nonsymbolic numerical tasks. These findings support the proposal that underlying nonsymbolic representations play a role in children’s mathematical development.     

The Numerical Stroop Effect in Primary School Children: A Comparison of Low,Normal, and High Achievers

Sixty-six primary school children were selected, of which 21 scored low on a standardized math achievement test, 23 were normal, and 22 high achievers. In a numerical Stroop experiment, children were asked to make numerical and physical size comparisons on digit pairs. The effects of congruity and numerical distance were determined. All children exhibited congruity and distance effects in the numerical comparison. In the physical comparison, children of all performance groups showed Stroop effects when the numerical distance between the digits was large but failed to show them when the distance was small. Numerical distance effects depended on the congruity condition, with a typical effect of distance in the congruent, and a reversed distance effect in the incongruent condition. Our results are hard to reconcile with theories that suggest that deficits in the automaticity of numerical processing can be related to differential math achievement levels. Immaturity in the precision of mappings between numbers and their numerical magnitudes might be better suited to explain the Stroop effects in children. However, as the results for the high achievers demonstrate, in addition to numerical processing capacity per se, domain-general functions might play a crucial role in Stroop performance, too.     

Individual differences in children's mathematical competence are related to the intentional but not automatic processing of Arabic numerals

In recent years, there has been an increasing focus on the role played by basic numerical magnitude processing in the typical and atypical development of mathematical skills. In this context, tasks measuring both the intentional and automatic processing of numerical magnitude have been employed to characterize how children’s representation and processing of numerical magnitude changes over developmental time. To date, however, there has been little effort to differentiate between different measures of ‘number sense’. The aim of the present study was to examine the relationship between automatic and intentional measures of magnitude processing as well as their relationships to individual differences in children’s mathematical achievement. A group of 119 children in 1st and 2nd grade were tested on the physical size congruity paradigm (automatic processing) as well as the number comparison paradigm to measure the ratio effect (intentional processing). The results reveal that measures of intentional and automatic processing are uncorrelated with one another, suggesting that these tasks tap into different levels of numerical magnitude processing in children. Furthermore, while children’s performance on the number comparison paradigm was found to correlate with their mathematical achievement scores, no such correlations could be obtained for any of the measures typically derived from the physical size congruity task. These findings therefore suggest that different tasks measuring ‘number sense’ tap into different levels of numerical magnitude representation that may be unrelated to one another and have differential predictive power for individual differences in mathematical achievement.     

Basic numerical skills in children with mathematics learning disabilities: a comparison of symbolic vs non-symbolic number magnitude processing

Forty-five children with mathematics learning disabilities, with and without comorbid reading disabilities, were compared to 45 normally achieving peers in tasks assessing basic numerical skills. Children with mathematics disabilities were only impaired when comparing Arabic digits (i.e., symbolic number magnitude) but not when comparing collections (i.e., non-symbolic number magnitude). Moreover, they automatically processed number magnitude when comparing the physical size of Arabic digits in an Stroop paradigm adapted for processing speed differences. Finally, no evidence was found for differential patterns of performance between MD and MD/RD children in these tasks. These findings suggest that children with mathematics learning disabilities have difficulty in accessing number magnitude from symbols rather than in processing numerosity per se.     

Relations of basic arithmetic and motor skills in deaf elementary school children

3 segregated groups of Arab and Jewish deaf children of CA 10;9 (n = 28) were compared with a group of hearing Arab first graders (CA = 6;10, n = 32) on tests of basic arithmetic, static balance control, and the ability to suppress synkinetic finger movements. The hearing-impaired performed as well on arithmetic tasks and on the tests of synkinetic control as their normal peers who were four years younger, while on static balance they were even inferior to the latter. Significant correlations were found between the basic arithmetic and motor skills, within the hearing as well as within the hearing-impaired groups; these remained significant even within the small subgroups of the latter. As these results cannot be accounted for by low intelligence and neurological disturbances, or by direct or indirect effects of deficient language development, the assumption is supported that some type of neurological immaturity, unrelated to hearing loss, interferes with the acquisition of numerical skills in deaf children.     

Relationships among accumulated lifetime life events,prosocial skills,and behavior problems in elementary school children

Relationships among life events, prosocial skills, and behavior problems for 91 first through fourth grade children were examined. The data were based on mothers' reports of their children's lifetime life events, current prosocial skills, and current behavior problems. Higher prosocial skills scores significantly predicted lower Externalizing behavior problem scores. Higher weighted life events scores significantly predicted higher Internalizing behavior problem scores. The weighted life event scores×prosocial skills scores interaction did not significantly predict either Externalizing or Internalizing behavior problem scores. The more parsimonious main effects conceptualization suggests that life events and prosocial skills are orthogonal in young, elementary school children.     

The predictive value of numerical magnitude comparison for individual differences in mathematics achievement

Although it has been proposed that the ability to compare numerical magnitudes is related to mathematics achievement, it is not clear whether this ability predicts individual differences in later mathematics achievement. The current study addressed this question in typically developing children by means of a longitudinal design that examined the relationship between a number comparison task assessed at the start of formal schooling (mean age = 6 years 4 months) and a general mathematics achievement test administered 1 year later. Our findings provide longitudinal evidence that the size of the individual’s distance effect, calculated on the basis of reaction times, was predictively related to mathematics achievement. Regression analyses showed that this association was independent of age, intellectual ability, and speed of number identification.     

Number comparison and number ordering as predictors of arithmetic performance in adults: Exploring the link between the two skills,and investigating the question of domain-specificity

Recent evidence has highlighted the important role that number-ordering skills play in arithmetic abilities, both in children and adults. In the current study, we demonstrated that number comparison and ordering skills were both significantly related to arithmetic performance in adults, and the effect size was greater in the case of ordering skills. Additionally, we found that the effect of number comparison skills on arithmetic performance was mediated by number-ordering skills. Moreover, performance on comparison and ordering tasks involving the months of the year was also strongly correlated with arithmetic skills, and participants displayed similar (canonical or reverse) distance effects on the comparison and ordering tasks involving months as when the tasks included numbers. This suggests that the processes responsible for the link between comparison and ordering skills and arithmetic performance are not specific to the domain of numbers. Finally, a factor analysis indicated that performance on comparison and ordering tasks loaded on a factor that included performance on a number line task and self-reported spatial thinking styles. These results substantially extend previous research on the role of order processing abilities in mental arithmetic.     

The development of strategy use in elementary school children: working memory and individual differences

The current study tested the development of working memory involvement in children's arithmetic strategy selection and strategy efficiency. To this end, an experiment in which the dual-task method and the choice/no-choice method were combined was administered to 10- to 12-year-olds. Working memory was needed in retrieval, transformation, and counting strategies, but the ratio between available working memory resources and arithmetic task demands changed across development. More frequent retrieval use, more efficient memory retrieval, and more efficient counting processes reduced the working memory requirements. Strategy efficiency and strategy selection were also modified by individual differences such as processing speed, arithmetic skill, gender, and math anxiety. Short-term memory capacity, in contrast, was not related to children's strategy selection or strategy efficiency.     

Behavioral coaching in the development of skills in football, gymnastics, and tennis

A review of the literature indicates that methods of skill acquisition based on the operant paradigm have been scientifically validated with many motor behaviors. However, these procedures have been limited to the use of positive reinforcement for correct performance when applied to the acquisition of complex sports skills in natural settings. To find complementary procedures to enhance skill acquisition, a coaching method involving several behavioral techniques was developed that focused on remediation of errors. This coaching method combined the following components: (1) systematic use of verbal instructions and feedback, (2) positive and negative reinforcement, (3) positive practice, and (4) time out. Three sports, football, gymnastics, and tennis, were selected to determine the effectiveness and generality of this behavioral coaching method. A total of 23 male and female subjects, ranging in age from 11 to 35, was included in this study. Baseline data were first collected for each sport under standard coaching conditions. Next, the behavioral coaching method was evaluated depending on the sport in either a multiple baseline or a reversal design. The behavioral dimensions selected were blocking in football; backward walkovers, front hand springs, and reverse kips in gymnastics; and the forehand, backhand, and serve in tennis. Behavioral coaching was immediately effective in increasing the correct execution of complex skills in all three sports. Gains of up to 10 times the baseline performance were achieved in each sport. In football, behavioral coaching resulted in an increase in correct blocking performance from a baseline average of 5% to 51.3%. Gymnasts' performances increased from baseline averages of 2.7% to 52.6% across the three skills. In tennis, standard coaching produced an average of 6% correct performance which increased dramatically under behavioral coaching to 57% across the three strokes. The success of the behavioral coaching package used here suggests that a technology of behavior may offer additional and complementary strategies to the acquisition of motor skills in the natural environment.     

Cognitive and linguistic skills in Swedish children with cochlear implants - measures of accuracy and latency as indicators of development

The purpose of the present study was to examine working memory (WM) capacity, lexical access and phonological skills in 19 children with cochlear implants (CI) (5;7-13;4 years of age) attending grades 0-2, 4, 5 and 6 and to compare their performance with 56 children with normal hearing. Their performance was also studied in relation to demographic factors. The findings indicate that children with CI had visuospatial WM capacities equivalent to the comparison group. They had lower performance levels on most of the other cognitive tests. Significant differences between the groups were not found in all grades and a number of children with CI performed within 1 SD of the mean of their respective grade-matched comparison group on most of the cognitive measures. The differences between the groups were particularly prominent in tasks of phonological WM. The results are discussed with respect to the effects of cochlear implants on cognitive development.     

The relationship between deliberate practice,play, and futsal in childhood and adolescence and the development of different decision-making skills in professional female soccer players

Soccer players’ ability to make efficient and quick decisions has gained more importance due to the increase in game speed in the last few years. Therefore, this study aimed to examine the relationship between the engagement in previous developmental activities in soccer and futsal with the quality and speed of decision-making skills in different phases of sport development of elite female soccer players. The sample comprised 77 elite Brazilian professional female soccer players. Players’ decision-making skills were assessed based on an objective video-based test – TacticUP®. We used a retrospective questionnaire to collect information about previous participation in different developmental activities. The results showed that engagement in deliberate practice in soccer and futsal, especially during childhood and early adolescence, is related to a better quality of offensive decision-making skills, although showing small to medium effect sizes. We highlight that deliberate practice in futsal is associated only with offensive decision-making skills with the ball and near the ball. In turn, engaging in deliberate play in soccer, mainly in childhood and early adolescence, is related to quicker offensive and defensive decision-making skills, showing mostly medium effect sizes. To the best of our knowledge, it was the first study to measure the relationship of decision-making speed with developmental activities in soccer. It is concluded that deliberate practice in soccer and deliberate play in soccer are associated with different dimensions of decision-making skills.     

Context and culture in the socialization and development of personal achievement values: Comparing Latino immigrant families,European American families,and elementary school teachers

We documented cross-cultural similarities and differences in values concerning personal achievement between Latino immigrant parents, a group of multiethnic teachers, and European American parents. We also explored intergenerational similarities and differences between parents and their fifth-grade children. The theoretical premise was that sociodemographic factors, such as education, drive cultural values, with more formal education associated with individualistic values and less formal education associated with collectivistic/familistic values. Responding to open-ended social dilemmas relevant to family life, Latino immigrant parents, averaging a fifth-grade education, responded more familistically than the more highly educated multiethnic teachers or European American parents. In contrast, no group differences in values showed up in situations where school practices do not directly impact family life. Intergenerational differences were few; but, in family-centered scenarios, European American fifth graders were significantly more collectivistic than European American parents, a finding that suggested the possibility that, in an individualistic culture, individualism is socialized with age.     

The development and effectiveness of memory strategies in kindergarten and elementary school: Findings from the Würzburg and Göttingen longitudinal memory studies

This article reports findings of the Würzburg and Göttingen Longitudinal Memory Studies, which focused on children's verbal memory development. The studies started with 102 (German) kindergarten children in Würzburg and 86 second-graders in Göttingen who were tested on various memory measures, including sort-recall, memory capacity, metamemory, and verbal IQ. Assessments were repeated nine times, with adjacent measurement points separated by 6-month time intervals. The main goals of the studies concerned the assessment of utilization deficiencies during the course of strategy development, the analysis of factors influencing the effectiveness of strategy use, and the occurrence of multiple strategy use. Findings confirmed the outcome of previous longitudinal studies in that semantic organization strategies were not always accompanied by superior recall, but utilization deficiency problems were comparably rare. Strategy effectiveness was shown as being affected mainly by memory capacity and by children's tendency to engage in multiple strategy use. However, children did not start to apply multiple strategies effectively until the end of Grade 2.     

The role of executive function and social‐emotional skills in the development of literacy and numeracy during preschool: a cross‐lagged longitudinal study

The majority of evidence on the interplay between academic and non‐academic skills comes from high‐income countries. The aim of this study was to examine the bidirectional associations between Ghanaian children's executive function, social‐emotional, literacy, and numeracy skills longitudinally. Children (N = 3,862; M age = 5.2 years at time 1) were assessed using direct assessment at three time points over the course of two school years. Controlling for earlier levels of the same skill, early executive function predicted higher subsequent literacy and numeracy skills, and early literacy and numeracy skills predicted higher subsequent executive function, indicating that the development of executive function and academic skills is inter‐related and complementary over time. Early literacy and numeracy predicted subsequent social‐emotional skills, but early social‐emotional skills did not predict subsequent literacy and numeracy skills. The findings provide longitudinal evidence on children's learning and development in West Africa and contribute to a global understanding of the relations between various developmental skills over time.